Let A be an expanding n n integer matrix with j det(A)j = m. A standard digit set D for A is any complete set of coset representatives for Z n =A(Z n). Associated to a given D is a set T(A; D), which is the attractor of an aane iterated function system, satisfying T = d2D (T + d). It is known that T(A; D) tiles R n by some subset of Z n. This paper proves that every standard digit set D gives a...

This paper presents a method for lossy coding of still pictures using the theory of iterated function systems (IFS) which is a standard description for deterministic fractals. The coding concept is based on a blockwise fractal approximation of the original image by contraction mappings of itself using affine transformations which are a special case of IFSs. Data reduction is achieved by exploit...

Two novel views are presented on the trapdoor channel. First, by deriving the underlying iterated function system (IFS), it is shown that the trapdoor channel with input blocks of length n can be regarded as the nth element of a sequence of shapes approximating a fractal. Second, an algorithm is presented that fully characterizes the trapdoor channel and resembles the recursion of generating al...

We study parabolic iterated function systems with overlaps on the real line. We show that if a d-parameter family of such systems satisfies a transversality condition, then for almost every parameter value the Hausdorff dimension of the limit set is the minimum of 1 and the least zero of the pressure function. Moreover, the local dimension of the exceptional set of parameters is estimated. If t...

An algorithm is presented which, given the parameters of an Iterated Function System (IFS) which uses affine maps, constructs a closed ball which completely contains the attractor set of the IFS. These bounding balls are almost always smaller than those computed by existing methods, and are sometimes much smaller. The algorithm is numerical in form, involving the optimisation of centre-point an...

In this paper we investigate aspects of rigidity and flexibility for conformal iterated function systems. For the case in which the systems are not essentially affine we show that two such systems are conformal equivalent if and only if in each of their Lyapunov spectra there exists at least one level set such that the corresponding Gibbs measures coincide. We then proceed by comparing this res...

Iterated function systems have been recently applied to the field of image coding. This paper exploits the fractal properties included in the coding and decoding schemes, in order to add useful tools for image processing. The first contribution consists of an improvement of the classical fractal zoom which allows, with a unique code, to increase the image resolution without loss in sharpness. T...

We establish properties of a new type of fractal which has partial self similarity at all scales. For any collection of iterated functions systems with an associated probability distribution and any positive integer V there is a corresponding class of V -variable fractal sets or measures with a natural probability distribution. These V -variable fractals can be obtained from the points on the a...

Using the theory of infinite iterated function systems, we show that the Julia set of any function of the type G = λ exp ◦F , λ ∈ C \ {0} and with F : C → Ĉ a non–constant elliptic function, has Hausdorff dimension two. However there exist elliptic functions F such that the Julia sets of the maps G = exp ◦F are nowhere dense in C.

Let A be an expanding n n integer matrix with j det A j m A standard digit set D for A is any complete set of coset representatives for Z A Z Associated to a given D is a set T A D which is the attractor of an a ne iterated function system satisfying T d D T d It is known that T A D tiles R n by some subset of Z This paper proves that every standard digit set D gives a set T A D which tiles R w...